The fractional step method for incompressible flow simulations requires repeatedly solving a pressure Poisson equation, a process that is computationally expensive due to the need to iteratively solve large-scale linear systems. To accelerate this critical step, we introduce HyDEA (Hybrid Deep lEarning and iterative methods for Accelerated solutions), a novel framework that synergistically combines deep learning with classical iterative solvers. HyDEA leverages the complementary strengths of a Deep Operator Network (DeepONet), which learns to predict large-scale solution features, and the robust conjugate gradient (CG) or preconditioned conjugate gradient (PCG) method, which corrects fine-scale errors. Within a line-search framework, the DeepONet predicts effective search directions to dramatically accelerate convergence for sparse, symmetric positive-definite systems, while the iterative solver ensures accuracy and robustness. The framework is naturally extended to flows involving solid structures via a decoupled immersed boundary projection method, which maintains the core linear system structure. A key innovation is that the DeepONet is trained solely on fabricated linear systems, not on flow-specific data. This approach grants the model inherent generalization ability across diverse geometric complexities, Reynolds numbers, and mesh resolutions without retraining. Benchmarks demonstrate that HyDEA achieves superior efficiency and accuracy compared to standard CG/PCG solvers for flows with no obstacles, single or multiple stationary obstacles, and a moving obstacle—all using a single set of fixed network weights. The framework also exhibits a promising super-resolution capability. HyDEA's combination of robustness, efficiency, and generalization positions it as a transformative solver for practical computational fluid dynamics challenges.